Eigenvalue estimates of positive integral operators with analytic kernels

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 20, No. 3, June 200, pp Indian Academy of Sciences Eigenvalue estimates of positive integral operators with analytic kernels YÜKSEL SOYKAN Department of Mathematics, Art and Science Faculty, Zonguldak Karaelmas University, 6700, Zonguldak, Turkey yuksel MS received 7 July 2008; revised 7 April 200 Abstract. In this paper, we exhibit canonical positive definite integral kernels associated with simply connected domains. We give lower bounds for the eigenvalues of the sums of such kernels. Keywords. Eigenvalue problems; special kernels; integral operators; Hardy spaces; Smirnov classes; closed analytic curves; Ahlfors regular curves.. Introduction Let be a simply connected domain (with an infinite complement) in the extended complex plane C = C ( ). Given a Riemann mapping function (RMF) ϕ for, i.e. a conformal map of onto the unit disc, define a function K on by K (ζ, z) = ϕ (ζ ) 2 ϕ (z) 2, ζ,z, (.) ϕ(ζ)ϕ(z) for either of the branches of ϕ 2. This function is independent of the choice of mapping function ϕ (see p. 40 of [9]). Little s work is on integral operators defined by restricting the function K to a square. Specifically, suppose J is a (bounded) closed real interval then we obtain a compact symmetric operator T on L 2 (J ) defined by T f(s)= K (s, t)f (t)dt, f L 2 (J ), s J. J This operator is always positive in the sense of operator theory, i.e. T f, f 0, for all f L 2 (J ) (see [9]). Simple interesting examples can be constructed from standard conformal maps; for instance, if = (and J (, )) we obtain K (s, t) = /( st), and if is the strip Im z <π/4 and ϕ(z) = tanh z we obtain K (s, t) = sech(s t). For more examples, see [8]. The main object of study in Little s work is the asymptotic behaviour of the eigenvalues λ n (T ) of T and related operators. These are always positive and we assume that they are arranged in decreasing order with repetitions to account for multiplicities. This work is most complete in the case when is symmetric about the real line R, as in the two examples 333

2 334 Yüksel Soykan above, for in Proposition 4, p. 28 of [8] it is shown how to construct a q,0 <q< such that λ n (T ) q n in the sense that λ n (T ) = O(q n ) and q n = O(λ n (T )). If we think of operators of the form T as standard examples then an obvious problem is to consider natural combinations of such operators (they will always be positive). In [9], Little considered distinct simply connected domains, 2,..., N each of which is bounded by a line or circle. He proves, for instance, that if J N i= i is a closed interval then λ n (T + ) λ n (T ), where T + is just the sum T + T 2 + +T N with kernel K (s, t) + K 2 (s, t) + +K N (s, t), and T is the integral operator on L 2 (J ) whose kernel is the pointwise product K (s, t)k 2 (s,t)...k N (s, t). In this paper we will continue in the spirit of [9]. Our main result is Theorem.. Let C = (C,C 2,C 3,...,C N ) be an N-tuple of closed distinct curves on the sphere C and suppose that for each i, i N,C i is a circle, a line { }, an ellipse, a parabola { } or a branch of a hyperbola { }. Let D be a complementary domain of N i= C i and suppose that D is simply connected and contains the closed interval J.IfD i is the complementary domain of C i which contains D and J then there exists some constant m>0 such that mt D T D + T D2 + +T DN in the operator sense. In particular, for all n 0, mλ n (T D ) λ n (T D + T D2 + +T DN ). We shall prove Theorem. in 4 and 6 using Theorems 4. and 4.2 of 4. Recall that a complementary domain of a closed F C is a maximal connected subset of C F, which must be a domain. Remark.2. From now on, we shall fix the notation as in Theorem.. 2. An outline of proof of Theorem. An outline of our method of proof of Theorem. is as follows. Suppose C is a simply connected domain. Then there is a canonical Hilbert space E 2 ( ) of analytic functions on. The precise definition of these spaces will be recalled in 3; so these spaces will be taken for granted for the moment. If contains our closed interval J then there is a natural restriction map S : E 2 ( ) L 2 (J ), S f(s)= f(s), f E 2 ( ), s J. It was shown in 3 of p of [9] that S is compact and that S S = T. That is, S S is the positive integral operator on L2 (J ) with kernel K. Similarly if, 2,..., N C, each containing J, are simply connected domains

3 Eigenvalue estimates of positive integral operators 335 and S + : E 2 ( i ) L 2 (J ) is given by S + (f,f 2,...,f N )(s) = S f (s) + S 2 f 2 (s) + +S N f N (s) = f (s) + f 2 (s) + +f N (s), f i E 2 ( i ), i N,s J, then (it was shown in p. 43 of [9] that) S + S + = T + T 2 + +T N is the compact positive integral operator on L 2 (J ) with kernel K D + K D2 + +K DN where T i is the integral operator on L 2 (J ) with kernel K i. To prove Theorem. it will be sufficient to find a continuous linear operator A: E 2 (D) N i= E2 (D i ) (see 4, Theorems 4. and 4.2). We shall be able to put m =. A 2 Suppose now that D is as in Theorem. and the operator A as in figure exists and is continuous. Then, for all f L 2 (J ), Hence Hence, by (2.) T D f, f = S D S D f, f = S D f 2 = A S + f 2 A 2 S + f 2 = A 2 S + S + f, f = A 2 (T D + T D2 + +T DN )f, f. T D A 2 (T D + T D2 + +T DN ). (2.) A 2 T D T D + T D2 + +T DN E 2 (D) S D A N i= E2 (D i ) S + L 2 (J ) Figure. Operator A.

4 336 Yüksel Soykan in the operator sense. In particular, for all n 0, A 2 λ n(t D ) λ n (T D + T D2 + +T DN ). This proves Theorem.. Although it takes a lot of effort to verify technical details, especially continuity, the operator A is quite easy to understand. The operator A comes from Cauchy s integral formula. The boundary D of D can be decomposed naturally as D = 2 N, where i C i. Cauchy s integral formula f(z)= N j= 2πi j f(ζ) ζ z dζ, f E2 (D), z D can be validated (see Theorem 3.2). The function f is the non-tangential limit function of f. In Theorem., it is insisted that the curves (C,C 2,C 3,...,C N ) are distinct. But once Theorem. has been proved this restriction can be removed. Suppose (C,C 2, C 3,...,C R ) are curves as in Theorem.. Suppose R>N, that is (C,C 2,C 3,...,C N ) are distinct and that each C i with i>nalready appears in the list (C,C 2,C 3,...,C N ). Clearly N R C i = C i, i= i= so that both sets of curves have the same complementary domains. Suppose D is one of them and that D is simply connected and contains J. Let k be a positive integer such that each C i appears at most k times in the list (C,C 2,C 3,...,C R ). Then so T D + T D2 + +T DN T D + T D2 + +T DR m(t D ) T D + T D2 + +T DR. 3. Preliminaries 3. Hardy and Smirnov classes We will need the theory of Hardy and Smirnov classes of analytic functions. The reader is referred to [5 7,,3] and references therein for a basic account of the subject. Here is a synopsis. If is a σ -rectifiable arc (see subsection 3.2 for definition) in C and f is integrable with respect to arc-length measure the notations f(z)dz, f(z) dz

5 Eigenvalue estimates of positive integral operators 337 will be used to denote, respectively, the complex line integral of f along and the integral of f with respect to arc-length measure. In the first case we assume has an orientation. The notation L p ( ) will denote the L p space of normalized arc length measure on with the factor 2π. T will be used to denote the boundary of the open unit disc. The space H ( ) is just the set of all bounded analytic functions on with the uniform norm. For p<, H p ( ) is the set of all functions f analytic on such that sup 0<r< 2π f(re iθ ) p dθ<. (3.) 2π 0 For p, H p (T) is defined as the set of all f L p (T) such that f(z)z m dz = 0, m = 0,, 2,... (3.2) 2πi T We have additional definitions of H 2 ( ) and H 2 (T). H 2 ( ) is the set of all analytic functions f on of the form f(z)= a n z n with a = (a n ) l 2 (3.3) n=0 and H 2 (T) is the set of all f L 2 (T) of the form f(e iθ ) = a n e niθ with a = (a n ) l 2. (3.4) n=0 For a more general simply-connected domain in the extended plane C = C ( ) and a conformal mapping ϕ from onto, a function g analytic in is said to belong to the Smirnov class E 2 ( ) if and only if g = (f ϕ)ϕ /2 for some f H 2 ( ) where ϕ /2 is an analytic branch of the square root of ϕ. The following theorem is proved in Duren [5]. Theorem 3.. Let be the inner domain of a rectifiable Jordan curve and suppose ψ: is a conformal equivalence. Then (a) ψ H ( ); (b) Every f E 2 ( ) has a nontangential limit function f L 2 ( ); (c) The map f f(e 2 ( ) L 2 ( )) is an isometric isomorphism onto a closed subspace E 2 ( ) of L 2 ( ), so f 2 = E 2 f 2 ( ) L 2 ( ) = f(z) 2 dz f E 2 ( ); 2π (d) For all f E 2 ( ) and z, f(z)= f(ζ) 2πi ζ z dζ.

6 338 Yüksel Soykan Proof. See pp of [5]. We need the following Theorem (see [2]) to show the existence of the operator A. When D and D i s are as in Theorem., the boundaries of D and D i s are reasonable and it will make sense for us to discuss the notion of a nontangential limit function and spaces E 2 ( D), E 2 ( D i ) analogous to H 2 ( ), H 2 (T). Theorem 3.2. Let D be as in Theorem.. Then (i) Every f E 2 (D) has a non-tangential limit function f L 2 ( D); (ii) Parseval s identity. The map f f is an isometric isomorphism of E 2 (D) onto a closed subspace E 2 ( D) of L 2 ( D), so f 2 = E 2 f 2 (D) L 2 ( D) = f(z) 2 dz, f E 2 (D); 2π (iii) Cauchy s Integral Formula. For all f E 2 (D) and z D, f(z)= f(ζ) 2πi D ζ z dζ. D 3.2 Arcs and curves An arc or closed curve is called σ -rectifiable if and only if it is a countable union of rectifiable arcs in C, together with ( ) in the case when. For instance, a parabola without is a σ -rectifiable arc, and a parabola with is a σ -rectifiable Jordan curve. Recall that a compact arc σ is called smooth if there exists some parametrization g: [a,b] σ such that g C [a,b] and g (t) 0, t [a,b]. Remark 3.3. Note that if σ is smooth then it is rectifiable, i.e., l(σ) = b a g (t) dt <. To define the arc length parametrization of σ put s = s(t) = t a g (u) du for a t b so that 0 s l(σ ). Then s (t) = g (t) and t s(t)([a,b] [0,l]) is C with strictly positive derivative. Hence also its inverse s t(s)([0,l] [a,b]) is C with strictly positive derivative. The arc length parametrization of the smooth arc σ is the map h:[0,l] σ satisfying h(s) = {the point on σ length s from the initial point (g(a))}, i.e., h(s) = g(t(s)), 0 s l. Since h (s) = g (t(s))t (s), h C [0,l] with non-zero derivative. Necessarily h (s) = since h (s(t)) = g (t)t (s) = g (t) s (t) = g (t) g (t). We need the following lemma.

7 Eigenvalue estimates of positive integral operators 339 Lemma 3.4. Let g C [0, ) with g (t) 0 (t 0). Suppose that c C and lim g(t) = c, (3.5) t lim t g (t) g = ω, (t) ω = (3.6) exist. Define σ = g([0, )) (c). Then (i) σ is a compact arc, (ii) σ is rectifiable, (iii) σ is smooth. Proof. The proof is straightforward and will be omitted. DEFINITION A simple σ -rectifiable arc C is said to satisfy hypothesis R if and only if, for some (hence all) a C, dz z a 2 <. Equivalently, if the function f(z)= z a belongs to L2 ( ). Lemma 3.5. Let be a simple, σ -rectifiable arc in C. Suppose that a C and that μ(z) = z a. Then satisfies hypothesis R if and only if μ( ) is rectifiable. Proof. Note that l(μ()) = μ dz (z) dz = z a 2 <. So the result is clear. Example. An analytic Jordan curve in C obviously satisfies hypothesis R. It is also easy to see that if satisfies hypothesis R so does μ( ), where μ is a linear equivalence: μ(z) = αz + β. Obviously R satisfies hypothesis R and therefore so does any line; and it can be easily seen that every parabola and hyperbola component satisfies hypothesis R. 4. The operator A and Cauchy integrals and easy cases The main result in this section is Theorem 4.. There exists a continuous linear operator A: E 2 (D) N i= E2 (D i ), say f (f,f 2,...f N ), such that f(z) f (z) + f 2 (z) + +f N (z) on D, for all f E 2 (D).

8 340 Yüksel Soykan Once this is proved our work will be completed, for we will then have T D A 2 (T D + T D2 + +T DN ), as proved in 3. Let j = D C j ( j N) then, by Cauchy s integral formula (Theorem 3.2(iii)), f(z)= 2πi N j= So the natural choice for f j is f j (z) = 2πi j Let F j be the extension of f j to C j : F j (z) = j f(ζ) ζ z dζ, f E2 (D), z D. f(ζ) ζ z dζ, z D j, j N. { f(z), if z j, 0, if z C j j. Of course, if j is finite or empty F j = 0. In general, F j (z) 2 dz = f(z) 2 dz 2π C j 2π j 2π D f(z) 2 dz = f 2 by Parseval s identity (see Theorem 3.2(ii)). So F j L 2 (C j ) and the map f F j is continuous. Hence Theorem 4. will be proved if the following continuity property of the Cauchy integral can be proved. Theorem 4.2. For each j( j N) the formula P j g(z) = g(ζ) 2πi ζ z dζ, g L2 (C j ), z D j (4.) C j defines a continuous linear operator P j mapping L 2 (C j ) into E 2 (D j ). The operator P j is a special case of a more general Cauchy integral operator or Cauchy transform. DEFINITION 2 Let C be a simple oriented σ -rectifiable arc satisfying hypothesis R. Forg L 2 ( ) the function C g on C is defined by C g(z) = g(ζ) 2πi ζ z dζ, g L2 ( ), z /. COROLLARY 4.3 If g L 2 ( ) then C g is analytic on C. If also is bounded and rectifiable then C g is analytic at and C g( ) = 0.

9 Eigenvalue estimates of positive integral operators 34 Proof. If z C, then by Schwarz s inequality 2π ( g(ζ) dζ g ζ z 2π ) dζ /2 ζ z 2 < +, by hypothesis R,soC g is defined on C. To prove that C g is analytic, let z C and choose an r>0 such that the open disc N 2r (z) centre z, radius 2r is contained in C. Let (h n ) be a sequence in C such that 0 < h n <r,for all n and h n 0. For each n, define I n = h n (C g(z + h n ) C g(z)). Then I n = 2πi = 2πi = 2πi g(ζ) h n ( ) dζ ζ z h n ζ z g(ζ) (ζ z h n )(ζ z) dζ g(ζ) ω(ζ) dζ, (4.2) (ζ z h n )(ζ z) where ω(ζ) is the unit tangent vector of at ζ ( ω(ζ) =a.e.). Now, for all n and almost all ζ, g(ζ)ω(ζ) (ζ z h n )(ζ z) g(ζ) r ζ z. Since the last function (of ζ )isinl ( ) it follows from Lebesgue s theorem, applied to (4.2), that (C g) (z) = lim I n = g(ζ) n 2πi (ζ z) 2 dζ. If is bounded then C,and if is rectifiable then L 2 ( ) L ( ). So if g L 2 ( ) then, for large values of z, C g(z) g(ζ) 2π z ζ dζ which tends to 0 as z, by Lebesgue s theorem. Therefore C g is analytic and 0 at. Remark 4.4. The orientation here is crucial and we must always be clear what it is before starting any manipulations. Obviously C g = C g. Recall that C j in Theorem 4.2 is oriented so as to wind positively around D and D j. The function P j g is then the restriction of C g to D j, where = C j. Lemma 4.5 (Invariance lemma). Let, 2 be simple oriented σ -rectifiable arcs satisfying hypothesis R, and suppose that, 2 are simply connected complementary domains of, 2, respectively. Suppose that μ is a fractional linear transformation such that

10 342 Yüksel Soykan μ( ) = 2 and μ( ) = 2 (with the same orientation). Then the map f C f defines a continuous linear operator mapping L 2 ( ) into E 2 ( ) if and only if the map g C 2 g 2 defines a continuous linear operator mapping L 2 ( 2 ) into E 2 ( 2 ), in which case the two operators are unitarily equivalent. Proof. Let us call the first map C and the second C 2 2 ; thus C f is the restriction to of the Cauchy integral C f and likewise for C 2 2. By symmetry, we need only prove one of the implications. So, suppose C 2 2 is a continuous linear operator mapping L 2 ( 2 ) into E 2 ( 2 ). Let V = V μ : L 2 ( ) L 2 ( 2 ) be the unitary operator Vf (ζ ) = f(μ (ζ ))μ (ζ ) /2, f L 2 ( ), ζ 2 and let U = U μ : E 2 ( 2 ) E 2 ( ) be the unitary operator Ug(z) = g(μ(z))μ (z) /2, g E 2 ( 2 ), z. Then UC 2 2 V : L 2 ( ) E 2 ( ) is a unitarily equivalent copy of C. To elucidate this operator, let f L 2 ( ) and z G 2, then C 2 2 Vf (z) = f(μ 2πi 2 (ζ ))μ (ζ ) /2 dζ ζ z = f (u)μ 2πi (u) /2 du, μ(u) z using the substitution ζ = μ(u). So if w, UC 2 2 Vf (w) = f (u) 2πi μ (u) /2 μ (w) /2 du. μ(u) μ(w) Now because μ is fractional linear it satisfies the functional equation μ (u) /2 μ (w) /2 μ(u) μ(w) = u w (4.3) as the reader will be able to verify. Hence C continuous and unitarily equivalent to C 2 2. = UC 2 2 V : L 2 ( ) E 2 ( ) is Lemma 4.6 (see [9]). Theorem 4.2 is true if C j is a circle or a line. Proof. We can map D j onto and C j onto the positively oriented unit circle using a fractional linear function. Thus in this case P j is unitarily equivalent to the operator C: L 2 (T) H 2 ( ): Cf (z) = f(ζ) 2πi T ζ z dζ, which is continuous, because if f(e iθ ) = a n e niθ n=

11 Eigenvalue estimates of positive integral operators 343 with a = (a n ) l 2 (Z), then Cf (z) = a n z n, n=0 z ; it is unitarily equivalent to the orthogonal projection of L 2 (T) onto H 2 (T). 5. David s theorem The remaining cases of Theorem 4.2 will be proved using David s theorem [0] (and Theorem 5.3 below). This has to do with the L 2 -space of a rectifiable Jordan curve. Suppose that is a positively oriented rectifiable Jordan curve with inner domain and outer domain G. We have two spaces of analytic functions E 2 ( ) on and E 2 (G) on G. Associated with these two we have closed subspaces E 2 ( ), E 2 ( G) of L 2 ( )( = = G) consisting of all nontangential limit functions of elements of E 2 ( ), E 2 (G), respectively. An obvious question arises Is L 2 ( ) = E 2 ( ) E 2 ( G), not necessarily orthogonal?. David s theorem gives a sufficient condition for the answer to be yes. The reason the question is obvious is that, when = T,E 2 ( ) = H 2 ( ) and so E 2 ( ) is all functions f on T of the form f(e iθ ) = a n e niθ n=0 with (a n ) l 2. And, because z z is an RMF for G = C, E 2 ( G) is all functions f on T of the form f(e iθ ) = b n e (n+)iθ n=0 with (b n ) l 2, so David s theorem is true when = T. Suppose now that our rectifiable Jordan curve does satisfy L 2 ( ) = E 2 ( ) E 2 ( G). It follows immediately from Banach s theorem that the (not necessarily orthogonal) projection (f,f 2 ) f (L 2 ( ) E 2 ( )) is continuous. It will be easy to show (Lemma 5.4(ii) below) that the Cauchy integral satisfies C (f + f 2 )(z) = C f (z), f E 2 ( ), f 2 E 2 ( G), z, so that, by (b) and (c) of Theorem 3., C (L 2 ( )) E 2 ( ). And if write f = f + f 2 as above we see that, for all f L 2 ( ), C f E 2 ( ) = C f E 2 ( ) = f L 2 ( ) M f L 2 ( ) for some M > 0, so that the Cauchy integral C defines a continuous linear operator mapping L 2 ( ) into E 2 ( ). There is a complete symmetry here and so C G defines a continuous linear operator mapping L 2 ( ) into E 2 (G) as well. Now the hypothesis of David s theorem involves the curve only. So in order to prove Theorem 4.2 in the case where C j is an ellipse it is sufficient to show that an ellipse satisfies the hypothesis of David s theorem, and because of the invariance Lemma 4.5, in order to verify Theorem 4.2 when C j is a parabola or hyperbola component it is sufficient to show that some fractional linear image μ(c j ) of C j satisfies the hypothesis of David s theorem. The key hypothesis in David s theorem is that the Jordan curve is Ahlfors regular.

12 344 Yüksel Soykan DEFINITION 3 An arc or closed curve is called Ahlfors regular if there is a constant k>0such that l(n r (z 0 ) ) kr, for all r>0 and z 0, where l is arc length measure. COROLLARY 5. If and k are as above then l(n r (z) ) 2kr, for all r>0 and z C. Proof. We only need to consider the case where N r (z). So choose z 0 N r (z). Then N r (z) N 2r (z 0 ). Hence the result. COROLLARY 5.2 If C is an Ahlfors regular Jordan curve and μ is a fractional linear function which is finite on then μ( ) is Ahlfors regular. Proof. Choose α>0 such that, for all z 0 and r 0 > 0, l(n r0 (z 0 ) ) αr 0. The pole a of μ is not on μ( ) so δ = inf ζ a > 0. ζ μ( ) Let K be the closure of the 2 δ neighbourhood of μ( ). Then M = sup μ (u) < +. u K If z μ( ) and w z < δ 2, then the line segment [z, w] K. Integrating μ along this arc shows that Hence also, if r< δ 2, Now μ (w) μ (z) M w z, z μ( ), w z < δ 2. μ (N r (z)) N Mr (μ (z)). N r (z) μ( ) μ(n Mr (μ (z)) ), so l(n r (z) μ( )) N Mr (μ (z)) μ (ζ ) dζ μ l(n Mr (μ (z)) ) ( μ αm)r, r < δ 2,

13 Eigenvalue estimates of positive integral operators 345 where μ is the uniform norm of μ on. Hence for all z μ( ) and r>0, ( l(n r (z) μ( )) max μ αm, 2l(μ()) ) r. δ Theorem 5.3 (David s theorem (Theorem 8, chapter 2 of [0]) or [4]). Let be an Ahlfors regular Jordan curve with inner domain and outer domain G. Let R be the set of all rational functions with no poles in and let R G be the set of all rational functions which vanish at and have no poles in Ḡ. Then L 2 ( ) = R R G where R, R G are the closures of R,R G respectively in L 2 ( ). To make use of David s theorem one more lemma is needed. It applies to a general rectifiable Jordan curve. Lemma 5.4. Let be a rectifiable Jordan curve with inner domain and outer domain G, and let R,R G be as in Theorem 5.3. (i) R E 2 ( ) and R G E 2 ( G). (ii) If f E 2 ( ) and f 2 E 2 ( G) then C f (z) 0 on G and C f 2 (z) 0 on. (iii) E 2 ( ) E 2 ( G) = (0). Proof. We show first that H ( ) E 2 ( ). Let g H ( ) and let ϕ: be an RMF for with inverse ψ:. By definition, g E 2 ( ) iff g = (f ϕ)ϕ /2 for some f H 2 ( ), equivalently iff (g ψ)ψ /2 H 2 ( ). But g ψ H ( ) and ψ /2 H 2 ( ) by Theorem 3.(a). Taking non-tangential limits gives R H ( ) E 2 ( ), so R E 2 ( ), because E 2 ( ) is closed in L 2 ( ). To prove that R G E 2 ( G) choose a and let X be the set of all functions on G of the form g(z) = f(z) with f H (G). z a It is sufficient to show that X E 2 ( G) because R G X. Choose a and let μ(z) = z a. Then μ( ) is a rectifiable Jordan curve with inner domain μ(g), so that H (μ(g)) E 2 (μ(g)), as above. Let U = U μ : E 2 (μ(g)) E 2 (G) be the unitary operator Uh(z) = h(μ(z))μ (z) /2, g E 2 (μ(g)), z G. The map h h μ is obviously an isomorphism of H (μ(g)) onto H (G), and μ (z) /2 is a multiple of z a.sox E2 (G) and (i) is proved.

14 346 Yüksel Soykan and To prove (ii) let f E 2 ( ) and z G. Let R(ζ) = ζ z, then R E2 ( ), by (i), C f (z) = f (ζ )R(ζ )dζ 2πi = f (ψ(u))r(ψ(u))ψ (u)du. 2πi T The last integrand here is in H (T), because it is the product of two elements of H 2 (T): (f ψ)ψ /2 and (R ψ)ψ /2. Hence C f (z) = 0 by eq. (3.2) with m = 0. The same argument shows that C f 2 0on. The proof of (iii) is more subtle. Suppose that m is a (finite) regular Borel measure with compact support X C. The Cauchy transform ˆm of m is defined by ˆm(z) = 2πi X dm(ζ ) ζ z. It is defined almost everywhere on C. The Cauchy transform is one of the main tools in the study of function algebras. One of its key properties is m = 0 if and only if ˆm(z) = 0, for almost all z C (5.) (see p. 55 of [3]). Now suppose f E 2 ( ) E 2 ( G) and let m be the measure on : dm(ζ ) = f(ζ)dζ = f(ζ)ω(ζ) dζ, where ω(ζ)is the unit tangent vector of at ζ. Clearly ˆm(z) = C f(z),for all z C. Now is rectifiable, so it has plane measure 0. It follows from (ii) that ˆm = 0 a.e. Therefore m = 0, so f = 0. Theorem 5.5. Let be an Ahlfors regular Jordan curve with inner domain and outer domain G. (i) L 2 ( ) = E 2 ( ) E 2 ( G). (ii) The map f C f is a continuous linear operator mapping L 2 ( ) into E 2 ( ) and the map f C f G is a continuous linear operator mapping L 2 ( ) into E 2 (G). Proof. Let R,R G be as before. Let f E 2 ( ). Then, by David s theorem f = g + h, where g R E 2 ( ) and h R G E 2 ( G). Therefore h = f g E 2 ( ) E 2 ( G) and so h = 0 by Lemma 5.4(iii). So E 2 ( ) = R and likewise E 2 ( G) = R G. Hence L 2 ( ) = E 2 ( ) E 2 ( G). Now, for f L 2 ( ), let us write f = f + f 2 with f E 2 ( ), f 2 E 2 ( G). By Banach s theorem the map f f, L 2 ( ) E 2 ( ) is continuous. By Lemma 5.4(ii) C f = C f.

15 Eigenvalue estimates of positive integral operators 347 By (b) and (c) of Theorem 3., C f E 2 ( ) and C f E 2 ( ) M f L 2 ( ) for some constant M > 0. If a and μ(z) = z a then μ(g) is the inner domain of μ( ), an Ahlfors regular curve, by Corollary 5.2. Our previous argument shows that f C μ( ) f μ(g) is a continuous linear operator mapping L 2 (μ( )) into E 2 (μ(g)). Hence, by the invariance Lemma 4.5, f C f G is a continuous linear operator mapping L 2 ( ) into E 2 (G). 6. Completion of the proof of Theorem 4.2 We shall find sufficient conditions for a Jordan curve to be Ahlfors regular and show that they apply to C j or to some fractional linear image μ(c j ) of C j (see Lemma 4.5). DEFINITION 4 A (simple) rectifiable arc with arc length parametrization h C [0,L] is called a Lavrentiev arc if there is an α>0such that Theorem 6.. h(s) h(t) α s t, s,t [0,L]. (i) A simple smooth arc is a Lavrentiev arc. (ii) A Lavrentiev arc is Ahlfors regular. (iii) If the arc is the union 2 n of Ahlfors regular arcs then itself is Ahlfors regular. (iv) A piecewise smooth Jordan curve is Ahlfors regular. Proof. Recall, from Remark 3.3, that is rectifiable with total length L, say, and that its arc length parametrization h C [0,L]. Now suppose is not a Lavrentiev arc. Then there are sequences (s n ), (t n ) [0,L] such that h(s n ) h(t n ) n s n t n, n. (6.) Assume that these two sequences are convergent, say s n s 0,t n t 0. It follows that h(s 0 ) h(t 0 ) = 0, so that s 0 = t 0. Now, for all n h(t n ) h(s n ) = tn s n h (u)du = (t n s n ) 0 (t n s n )h (t 0 ), h (s n + v(t n s n ))dv by Lebesgue s bounded convergence theorem. But h (t 0 ) =, contradicting (6.).

16 348 Yüksel Soykan To prove (ii) let z 0 = h(t),with t [0,L]. If z = h(s) is a general point of then, for r>0, z N r (z 0 ) h(s) h(t) <r s t < r α ( s t r α,t + r ). α Remembering that s is arc length we see that ( l(n r (z 0 ) ) arc length from h t r ) α ( to h t + r ) α = 2 α r. To prove (iii) suppose that = 2 n where each i is Ahlfors regular, say l(n r (z i ) i ) k i r, z i i,r >0. (6.2) Let z 0, say z 0 i. Then we certainly have and by Corollary 5., Varying i gives l(n r (z 0 ) i ) k i r, r > 0 l(n r (z 0 ) j ) 2k j r, j i. l(n r (z 0 ) ) 2(k + k 2 + +k n )r, z 0,r > 0. Item (iv) is now clear. COROLLARY 6.2 If C j is an ellipse then Theorem 4.2 is true. Now suppose C j is a parabola. By the invariance Lemma 4.5, we can assume that C j is the parabola y 2 = 4( x). Now let μ(z) = z. Another application of the same lemma and Theorem 6.(iv) shows that it is sufficient to show that μ(c j ) is piecewise smooth. Now μ(c j ) has a cusp at 0 = μ( ). But we can show that the parts of μ(c j ) in the upper and lower half planes are smooth. The upper part of μ(c i ) has parametric function g C [0, ), g(t) = ( + it) 2, t 0, so that g (t) = 2i.Nowg(t) 0ast + and (+it) 3 g (t) + it 3 i g = i (t) ( + it) 3 = i + 3 t ( ) i + 3 t

17 Eigenvalue estimates of positive integral operators 349 as t +. So by Lemma 3.4, g[0, ) g( ) is smooth and, similarly g(, 0] g( ) is smooth. Finally, let C j be a hyperbola component, say C j = sin(α + ir), where 0 <α< π 2. Again put μ(z) = z. The upper part of μ(c j ) is parametrized by g(t) = Again g(t) 0ast + and g (t) g (t) sin α cosh t + i cos α sinh t, t 0. α sinh t + i cos α cosh t) sin α cosh t + i cos α sinh t 2 = (sin sin α sinh t + i cos α cosh t (sin α cosh t + i cos α sinh t) 2 (sin α tanh t + i cos α) sin α coth t + i cos α 2 = sin α tanh t + i cos α (sin α coth t + i cos α) 2 ie iα (ie iα ) 2 = ieiα. Arguing as before we see that μ(c j ) is piecewise smooth. Remark 6.3. Theorem. creates the impression that an important role is to be played by the very specific curves mentioned; but on closer inspection of proof it can be seen that the argument runs on the lines conic section piecewise smooth Lavrentief arc Ahlfors regular and then Ahlfors regularity is used to prove continuity of the mapping A. 7. Conclusions and further work In this paper we proved Theorem. in detail by showing the continuity of the operator A which is followed by the continuity of the Cauchy integral operators (Theorem 4.2). Also we need Theorem 3.2 which shows that the operator A exists. There are some difficulties in generalizing Theorem. for more general case than conic sections. We raise some of them in the following questions: Under what conditions on an N-tuple C = (C,C 2,C 3,...,C N ) of closed distinct curves in the extended plane (a) are the results of Theorem 3.2 true? (so that the operator A exists) and (b) is Theorem 4.2 true?, i.e., is each Cauchy integral operators continuous? (i), (ii) and (iii) of Theorem 3.2 are consequences of being ψ H ( ) for the bounded ψ case of D and H ( ) for the unbounded case of D where a C\ D (see (ψ a) 2 Theorems 25 and 30 of [2]]. Arsove [,2] gives a complete description of bounded simply connected domains for which the derivative of a conformal mapping of the unit disk onto belongs to H. In fact, according to Theorem of [], if is a bounded simply connected domain and χ is a Riemann mapping function for then χ is in the Hardy class

18 350 Yüksel Soykan H if and only if can be parametrized as a rectifiable closed curve. This description shows that there is no real need to restrict oneself to domains bounded by pieces of quadrics so that Theorem 3.2 can be improved. Therefore, it seems that the more difficult part of future work is to prove the continuity of the Cauchy integral operators (Theorem 4.2). Further work includes investigating more complex scenarios such as choosing all closed curves as analytic σ -rectifiable Jordan curve. Acknowledgement This work is based in part on the author s doctoral thesis (University of Manchester), written under the direction of Dr Graham Little. The author expresses his gratitude to the referee for carefully reading the manuscript and for providing several valuable pointers which improved it. References [] Arsove M G, Some boundary properties of the Riemann mapping function, Proc. Am. Math. Soc. 9(3) (968) [2] Arsove M G, A correction to Some boundary properties of the Riemann mapping function, Proc. AMS 22(3) (969) 7 72 [3] Browder A, Introduction to Function Algebras (New York, Amsterdam: W.A.Benjamin Inc.) (969) [4] David J, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. E.N.S. 7 (984) [5] Duren P L, Theory of H p spaces (New York: Academic Press) (970) [6] Goluzin G M, Functions of a Complex Variable (Amer. Math. Soc., Providence, RI) (969) (translated from Russian) [7] Khavinson D, Factorization theorems for certain classes of analytic functions in multiply connected domains, Pacific. J. Math. 08 (983) [8] Little G, Asymptotic estimates of the eigenvalues of certain positive Fredholm operators, Math. Proc. Cambridge Philos. Soc. 9 (982) [9] Little G, Equivalences of positive integral operators with rational kernels, Proc. London. Math. Soc. (3) 62 (99) [0] Meyer Y and Coifman R, Wavelets, Calderon-Zygmund and multilinear operators Cambridge Studies in Advanced Mathematics 48, Translation of vols 2 3 of Ondelettes et operateurs by D H Salinger (997) [] Privalov I I, Boundary Properties of Analytic Functions (Moscow) (950) (Russian, German translation: Randeigenschaften analytischer Funktionen, Deutscher Verlag der Wiss., Berlin (956)) [2] Soykan Y, Boundary behaviour of analytic curves, Int. J. Math. Anal. (3) (2007) 33 57, index.html [3] Soykan Y, On equivalent characterisation of elements of Hardy and Smirnov spaces, Int. Math. Forum 2(24) (2007) 85 9,